Topological magnetic textures in magnetic topological insulators

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Topological magnetic textures in magnetic topological insulators

Nisarga Paul and Liang Fu

Phys. Rev. Research 3, 033173 – Published 20 August 2021

Abstract

The surfaces of intrinsic magnetic topological insulators (TIs) host magnetic moments exchange-coupled to Dirac electrons. We study the magnetic phases arising from tuning the electron density using variational and exact diagonalization approaches. In the dilute limit, we find that magnetic skyrmions are formed, which bind to electrons, leading to a skyrmion Wigner crystal phase while at higher densities spin spirals accompanied by chiral one-dimensional channels of electrons are formed. The binding of electrons to textures raises the possibility of manipulating textures with electrostatic gating. We determine the phase diagram capturing the competition of intrinsic spin-spin interactions and carrier density and comment on the possible application to experiments in magnetic TIs and spintronic devices such as skyrmion-based memory.

Received 10 March 2021
Accepted 7 July 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.033173

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Skyrmions Spin texture

Topological insulators

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Nisarga Paul^1,2 and Liang Fu²

¹Department of Physics, Harvard University, Cambridge, Massachusetts, USA
²Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

Twisted magnetic topological insulators

Chao-Kai Li, Xu-Ping Yao, and Gang Chen

Phys. Rev. Research 3, 033156 (2021)

Article Text

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Vol. 3, Iss. 3 — August - October 2021

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Images

Figure 1
(a) Top: Illustration of an intrinsic magnetic TI. In the case of ${MnBi}_{2} {Te}_{4}$ the alternating layers consist of the TI ${Bi}_{2} {Te}_{3}$ and magnetic layer MnTe. The magnetic layer can host real-space topological textures such as Néel skyrmions, shown in (b). Bottom: Spectrum of Dirac electrons coupled to Néel skyrmion texture with radius $b = 10$ lattice spacings and shape parameters $α = 3 / 2, β = 1 / 2$ [Eq. (5)] plotted against total (spin $+$ orbital) angular momentum, showing a branch of chiral midgap bound states. (b) Top: Néel skyrmion textures with radii $b = 1, 5,$ and 10 lattice spacings and the same shape parameters. Bottom: Magnitude of spin-up component of the smallest $| E |$ Dirac bound state in the presence of the corresponding texture.
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Figure 2
Left: Skyrmion profiles $θ (r)$ , where $θ = 0$ corresponds to $+ \hat{z}$ , for the shape parameters shown [cf. Eq. (5)]. Right: The cutoff $k_{UV}$ below which a skyrmion of the given radius becomes favorable upon doping a single electron. (Distances are measured in lattice spacings.)
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Figure 3
(a) Ground-state (g.s.) sinusoidal Néel spiral of $H_{e} + H_{e S}$ , characterized by wave vector $q$ in the $x$ direction, as a function of doping. The slope $\sim 2 k_{F}$ for the underlying dispersion. A $10^{3} \times 500$ lattice with unit spacing and cutoff $k_{UV} = π / 20 a$ was used. (b) Spectrum along the red line of the inset mini Brillouin zone (BZ) for Dirac electrons in a sinusoidal Néel spiral of wavelength $m a$ in the $x$ direction (here, $m = 20, J = 0.5$ ). Conduction band states descend to form midgap bound states localized in the $x$ direction. Localization leads to nearly flat bands in the $k_{x}$ direction, with dispersion tuned by $J$ . (c) Magnitude of spin-up component of $E = 0$ Dirac bound states in the presence of the spin spiral above. The velocities are indicated by long arrows.
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Figure 4
The largest eigenvalue $χ_{e S} (q)$ of the spin susceptibility defined in Eq. (11). The dominant peak shifts discontinuously away from $q = 0$ at a critical $k_{F} a \approx 0.54$ . The transition occurs at a point in the Brillouin zone where a Dirac cone is a good approximation to the dispersion of the $10^{3} \times 500$ lattice model ( $a = 1$ ). Left and right insets display the inter- and intraband spin susceptibilities, respectively. The $2 k_{F}$ peak is entirely due to the latter, and the UV dependence is captured entirely in the former.
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Figure 5
Phase diagram describing Dirac electrons coupled to spins as the electron density and DM interaction are varied [within $(Δ d, Δ k_{F} a) = (0.15, 0.04)$ ] with $A = K = 0.1$ , $J = 0.5$ , and $a = 1$ on a $100 \times 100$ lattice. The spins exhibit stripe orders at wave vectors $q = 0$ , $2 k_{F},$ and $q_{0}$ [defined in Eq. (20)], as determined by the spin susceptibility peak.
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